Write an equation for each translation. See Problem 5.

36. y = sin x , π  units to the left y equals sine x comma pi . unitstotheleft
37. y = cos x , π 2 units down y equals cosine x comma , fraction pi , over 2 end fraction . unitsdown
38. y = sin x , 3  units up y equals sine x comma 3 . unitsup
39. y = cos x , 1.5  units to the right y equals cosine x comma 1.5 . unitstotheright

40. Temperature The table below shows water temperatures at a buoy in the Gulf of Mexico on several days of the year. See Problem 6.

    Day of Year	16	47	75	106	136	167	198	228	258	289	319	350
    Temperature (°F)	71	69	70	73	77	82	85	86	84	82	78	74

    1. Plot the data.
    2. Write a cosine model for the data.

B Apply

Write an equation for each translation.

41. y = cos x , 3 y equals cosine x comma 3  units to the left and π units up
42. y = sin x ,  π 2 y equals sine x comma , pi over 2  units to the right and 3.5 units up
43. Think About a Plan The function y = 1.5 sin π 6 ( x − 6 ) + 2 y equals 1.5 sine , pi over 6 . open , x minus 6 , close . plus 2  represents the average monthly rainfall for a town in central Florida, where x represents the number of the month (January = 1, February = 2, and so on). Rewrite the function using a cosine model.
    • How does the graph of y = sin x y equals sine x  translate to the graph of y = cos x ? y equals cosine x question mark
    • What parts of the sine function will stay the same? What must change?

Write a cosine function for each graph. Then write a sine function for each graph.

44. 
45. 

46. The graphs of y = sin x  and  y = cos x y equals sine x , and , y equals cosine x  are shown below.

    

    1. What phase shift will translate the cosine graph onto the sine graph? Write your answer as an equation in the form sin x = cos ( x − h ) . sine x equals cosine open x minus h close .
    2. What phase shift will translate the sine graph onto the cosine graph? Write your answer as an equation in the form cos x = sin ( x − h ) . cosine x equals sine open x minus h close .
47. 1. Open-Ended Draw a periodic function. Find its amplitude and period. Then sketch a translation of your function 3 units down and 4 units to the left.
    2. Reasoning Suppose your original function is f ( x ) . f open x close .  Describe your translation using the form g ( x ) = f ( x − h ) + k . g open x close equals f open x minus h close plus k .
48. 1. Write y = 3 sin ( 2 x − 4 ) + 1 y equals 3 sine open 2 x minus 4 close plus 1  in the form y = a sin b ( x − h ) + k . y equals eh sine b open x minus h close plus k .  (Hint: Factor where possible.)
    2. Find the amplitude and period. Describe any translations.

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